# Talk:Euler's theorem

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Field: Number theory

## Math investigatory project

What is math investigatory project?? —Preceding unsigned comment added by 210.5.83.81 (talkcontribs) 07:19, 5 November 2004

Such a statement has now been removed from the article.--Leif edling (talk) 06:17, 18 May 2009 (UTC)

## F+V=E+2?

I was taught that Euler's Theorem stated, that, in solids, the number of faces (F) plus the number of vertices (V)equals the number of edges (E) plus two. Is that not Euler's Theorem? —Preceding unsigned comment added by 207.225.245.182 (talk) 00:33, 29 May 2009 (UTC)

There are many mathematical results named after Euler. The one you mention is sometimes called Euler's formula, and is generalised in the Euler characteristic. Gandalf61 (talk) 09:52, 29 May 2009 (UTC)

## Statement

This theorem is stated as an implication: if a and n are coprime, then we know that the value ${\displaystyle a^{\varphi (n)}{\bmod {n}}}$ is 1. That's the way the theorem is stated here and in most textbooks (I just checked Fundamental number theory with applications by Richard A. Mollin, page 93). This is the non-trivial result.

The converse is true, but it's so trivial that it does not have to be stated. An equivalent form of the converse is: if a and n are not coprime, then we know that the value ${\displaystyle a^{\varphi (n)}{\bmod {n}}}$ is not 1. This is obvious: Let p be any prime that divides both a and n, then for all positive integers x we know that p divides ${\displaystyle a^{x}{\bmod {n}}}$, hence this value is not 1.

Please keep the statement as commonly used. The converse is barely worth a note somewhere in the article. Misof (talk) 08:30, 18 March 2011 (UTC)

I completely agree. The article must give a statement of the theorem as it appears in most reliable sources, which is as a one-way implication. I don't mind whether we also mention the converse or not - as Misof says, it is trivially true. But we must not change the statement of the original theorem. Gandalf61 (talk) 09:15, 18 March 2011 (UTC)

## History of Euler's theorem

The article states that Euler's theorem was first proved in 1736. That can't be correct because according to Wikipedia's article "Euler's totient function", φ (n) was not defined by Euler until 1760. What Euler proved in 1736 was Fermat's little theorem. (As usual, Fermat presented the theorem but not its proof.) Even in 1760, Euler did not use the symbol φ (n) for the totient function; Gauss did that later. Also, Euler didn't call φ (n) "the totient function"; instead, he called it "numerus partium ad N primarum" (i.e., the number of parts prime to N -- or the number of natural numbers that are smaller than N and relatively prime to N). Euler's proof of "Euler's theorem" first appeared in print in 1763. I'll make appropriate changes to the article. Cwkmail (talk) 19:29, 20 December 2012 (UTC)

## Problem with proof 1?

Proof 1 says: "If a is any number coprime to n then a is in one of these residue classes, and its powers a, a^2, ..., a^k ≡ 1 (mod n) are a subgroup."

But {a, a^2, ..., a^k} cannot be a subgroup without a multiplicative identity element; I believe it should instead read "and its powers a^0, a^1, a^2, ..., a^k". — Preceding unsigned comment added by 73.179.121.212 (talk) 01:08, 19 November 2015 (UTC)

Note: I tried deleting this because my notion was wrong (the proof is correct). For some reason Oshwah reverted my deletion. In any case, the proof is correct because a^k (mod n) forms the multiplicative identity element, so {a, a^2, ..., a^k} is indeed a subgroup. — Preceding unsigned comment added by 73.179.121.212 (talk) 16:00, 19 November 2015 (UTC)

Euler’s theorem is not the basis for the RSA encryption algorithm. For r, s (> 1) such that x ^ rs = x mod a for all x it is necessary that a be square-free. We do not need r and s to be inverse in the ring of integers modulo Euler’s totient function, which, in the square-free case is the product of p – 1 as p ranges over the factors of a; it is sufficient for r and s to be inverse in the ring of integers modulo the smallest positive number which is divisible by p – 1 for each prime factor p. Further, we do not need Euler’s theorem to prove this; Fermat’s theorem (referred to in the article as Fermat’s “little” theorem) is sufficient.

Bukovets (talk) 14:15, 22 July 2016 (UTC)